This project entails the development and application of transformative tools for the design and modeling of imperfect and amorphous materials for photovoltaic (PV) applications. In particular, the work will focus on the fundamental connections between random matrix theory (RMT) and electronic structure theory, using the resulting links to leverage large scale simulations of disordered PV materials. Specifically, since one is in the end most interested in the distribution of eigenvalues, the question naturally arises if it is possible to calculate these distributions directly from the ensemble of structures. Surprisingly, in many cases, RMT allows the calculation of universal eigenvalue distributions from an apparently random distribution of matrices, without ever actually doing any matrix algebra. The work will be organized around four specific aims: 1) The construction of large data sets on amorphous silicon and organic bulk heterojunction PV that can be used to test the predictions of RMT; 2) Exploring the range of RMT models in the context of PV, pushing the fundamental limits of this technique; 3) Using RMT to constrain reduced models of extended systems, focusing on variables that rapidly approach the asymptotic predictions of RMT; 4) Leveraging DFT and RMT to describe amorphous group IV, III/V and II/VI semiconductors.

One of the grand scientific challenges of the 21st century is the development of a technology that can convert sunlight to electricity on a large scale at reasonable cost. Most of the proposed developments toward this end rely heavily on materials containing defects or inherently disordered materials as the active element. These materials have lower manufacturing costs, but typically also lower energy efficiency, which offsets their value in the solar marketplace. This work will develop mathematical and computational tools that will yield a deeper understanding of new solar technologies and accelerate the design of better conversion devices. The project will contribute to reducing our national carbon footprint and minimizing our dependence on foreign sources of fossil fuels. The work will also bring novel high-performance computing tools to bear on solar energy problems. Finally, through the training of students and postdocs, the project will help foster the next generation of scientists and engineers, ready to address problems relevant to society.

Project Report

Beta ensembles: Generalizing the notion of random matrices to arbitrary beta has led to new sampling methods for computing the eigenvalue distribution of the beta-Wishart ensemble as well as the singular values of the beta-MANOVA ensemble with general covariance. Isotropic entanglement. We developed a new method for computing the eigenvalue distribution of quantum many-body spin systems with generic interactions, which are of general interest for condensed matter physics and quantum computing applications. In this work, we found that the eigenvalue distribution (as described by the bandgaps) can be computed extremely accurately using only information from the first four moments of the distribution, producing a result that is far superior to standard techniques of approximating statistical distributions using the first four moments such as the Pearson and Gram-Charlier series. The key result is the "slider theorem", which computes a single nonadjustable parameter based on the discrepancies between classical and free convolutions, and allows for a highly accurate interpolation between these two limits. Free probability for Schrödinger operators. We developed a new approximation method for computing the band structures of disordered materials inspired by ideas behind isotropic entanglement. The approximation method uses techniques from free probability theory, previously regarded as a field of pure mathematics with no known applications outside of mathematics, and uses them to compute the eigenvalues of spectral properties of Schrödinger operators very accurately, without exact diagonalization. We also developed a new error analysis technique using Edgeworth expansions to quantify the discrepancy between the approximate and exact bandgaps. The error analysis can be computed a posteriori without knowing the exact statistical distribution of eigenvalues, allowing users to understand the errors inherent in the computation. We found the one of these highly accurate approximations was related to an established technique in condensed matter physics known as the coherent potential approximation. Hole traps in disordered silicon. A key challenge in engineering the band structure of silicon-based semiconductors is understanding how the electronic structure is affected by impurities such as hydrogen atoms which are inevitably introduced as part of the manufacturing process. Using genetic programming techniques, we sampled a large number of structures of nanocrystalline silicon that exhibit hole traps. Analyzing the resulting structures showed that hole traps are most strongly associated with unusual valences such as bridging hydrogen atoms and fivefold coordinated silicon atoms.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Henry Warchall
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Massachusetts Institute of Technology
United States
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